### Never an integer?

Consider the sequence defined by:                   (2N+K)!U[N] = Sum [ ------------------ ] K = 0,1,..,N              (2N-2K)!*(3K+1)!It starts like this:U[1] = 5/4 U[2] = 51/14U[3] = 277/20, U[4] = 1497/26It seems that all elements are rationals.Actually, some elements are integers.What is the 1st N for which U[N] is an integer?What is the value of U[N] modulo 10^50? (It's a BIG number)Answer format: Index,ValueHint:Consider the sequence V[N] = (6N+2)*U[N] and show it satisfies a recurrence relation:V[N+3] = A*V[N] + B*V[N+1] + C*V[N+2] (A,B,C integers)Warning:Finding the answer will be quite demanding for your machine ...
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Problem Maintainer :  Philippe_57721
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